Magnetic susceptibility of a paramagnetic material by Quincke’s

method

Objective

1. To determine the magnetic susceptibility χ of a given paramagnetic solution for a specific

concentration.

2. Calculate mass susceptibility χ′, Molar susceptibility χ″, Curie constant C and Magnetic

dipole moment.

Theory

When a material is placed within a magnetic field, the magnetic forces of the material's

electrons will be affected. This effect is known as Faraday's Law of Magnetic Induction.

However, materials can react quite differently to the presence of an external magnetic field.

This reaction is dependent on a number of factors, such as the atomic and molecular structure

of the material, and the net magnetic field associated with the atoms. The magnetic moments

associated with atoms have three origins. These are the electron motion, the change in motion

caused by an external magnetic field, and the spin of the electrons. In most atoms, electrons

occur in pairs with spins in opposite directions. These opposite spins cause their magnetic

fields to cancel each other. Therefore, no net magnetic field exists. Alternately, materials with

some unpaired electrons will have a net magnetic field and will react more to an external

field. Most materials can be classified as diamagnetic, paramagnetic or ferromagnetic.

Although you might expect the determination of electromagnetic quantities such as

susceptibility to involve only electrical and magnetic measurements, this practical shows how

very simple measurements of mechanical phenomena, such as the displacement of a liquid

column can be used instead. Quincke devised a simple method to determine the magnetic

susceptibility, χ, of a paramagnetic solution by observing how the liquid rises up between the

two pole pieces of an electromagnet, when a direct current is passed through the

electromagnet coil windings. A material’s magnetic susceptibility tells us how “susceptible”

it is to becoming temporarily magnetised by an applied magnetic field and defined as the

magnetization (M) produced per unit magnetic field (H).

χ = M/H (1)

Consider a paramagnetic medium in the presence of a uniform applied flux density Bo.

Loosely speaking, paramagnets are materials which are attracted to magnets. They contain

microscopic magnetic dipoles of magnetic dipole moment m which are randomly oriented.

However, in the presence of a uniform field B each dipole possesses a magnetic potential

energy U = −m• B, [1]. So they all tend to align up parallel to B, which is the orientation in

which their potential energy is minimum (i.e. most negative). Consequently, the liquid, which

contains many such dipoles, will tend to be drawn into the region of maximum field since this

will minimize its total magnetic potential energy. In other words, the liquid experiences an

attractive magnetic force Fm pulling it into the region of strongest field. The dipoles in the

liquid, FeCl3 solution for this experiment, are due to Fe3+

ions which are paramagnetic in

their ground-state. The “spins” of several outer electrons are aligned parallel to each other to

gives rise to a net magnetic moment m which is not compensated by other electrons.

A region of empty space permeated by a magnetic field H possesses an energy whose density

(energy per unit volume) is u = ½μoH2 [1], where μo is the magnetic permeability of vacuum.

In presence of a medium, this magnetic energy density may be written:

2

2

1Hu µ= (2)

where μ is the magnetic permeability of the medium and H = |H|. For fields which are not too

large, the magnetic permeability μ of a paramagnet can be treated as independent of the

applied field; i.e. it is a “constant”. Note that μ>μo for a paramagnet. The H vector has the

very useful property that its tangential component is continuous across a boundary, so that the

value of H in the air above the meniscus is equal to that in the liquid. This is in contrast to the

flux density, where B0 in air is different (less, in this case) from the value B in the liquid:

( )χµµµ +

===100

0 BBBH (3)

Suppose that, when the field is turned on, the meniscus in the narrow tube rises by an amount

h, relative to its zero-field position (see Fig. 1). A volume πr2h of air in the narrow tube (with

permeability μo) is, therefore, replaced by liquid. Hence, the magnetic potential energy of this

volume of space increases by an amount:

( )h)(2

1 22 rHU a πµµ ×−=∆ (4)

The work done by the upward magnetic force Fm in raising the liquid by an amount h is

22)(2

1

hrH

UF am πµµ −=

∆= (5)

When the liquid in one arm of the tube rises by h, it falls on the other arm by h. It continues

to rise till the upward magnetic force is balanced by the weight of the head of liquid. The

downward gravitational force on the head of liquid, of mass m, is given by

hgrmgFg ρπ 22== (6)

where ρ is the density of the liquid. However, there is also a very small additional upwards

force on the liquid due to the buoyancy of the air, which, strictly, ought to be included (By

the Principle of Archimedes, bodies immersed in any fluid, even air, experience this

buoyancy; you are yourself very slightly lighter by virtue of the surrounding air, though this

effect is extremely tiny compared to that which you experience when immersed in a much

denser fluid, such as water). The liquid in the narrow column displaces a volume of air, while

that in the wide column is replaced by air, and this leads to a net upwards buoyancy force on

the narrow column given by

hgrF ab ρπ 22= (7)

where ρa is the density of the air. Combining all these forces, we have Fm =Fg -Fb, so that

( ) hgrrH aa )(22

1 222 ρρππµµ −=− (8)

Using Eq. 3 in Eq. 8, we finally obtain the volume susceptibility, which is a dimensionless

quantity:

[ ] aaoB

hg χρρµχ −−=

2)(4 (9)

Where aχ is the susceptibility of air. In practice, the corrections due to air are negligible.

There will also be a small but significant diamagnetic (i.e. negative) contribution to the

susceptibility mainly due to water. The total susceptibility of the solution is given by χ = χFe

+ χwater. In the present work you will correct χ to yield the true value of χFe. Some other

parameters are defined in terms of volume susceptibility as follows:

Mass Susceptibility is given by: χ′ = χ/ρ (10)

Molar Susceptibility is given by: χ′′ = χ′ M (11)

where M= Molecular weight

Curie constant is given by: C = χ′′/T (12)

where T= Temperature of sample

Magnetic moment µ of dipole of the specimen by relation

µ = 2.8241√C (13)

where µ is expressed in Bohr magnetron µB with a value of 9.272 × 10-24

A-m2

Literature value of molar susceptibility of FeCl3 is +1.69 x 10-8 m3/mol [2].

Since the thermal effects tend to destroy the alignment of magnetic dipoles, so the susceptibility

of a paramagnet decreases as the temperature T is increased. Using statistical mechanics, it may

be shown that at high temperatures (kT >> mB) the contribution χFe of the paramagnetic Fe3+ ions

to the volume susceptibility of the solution is given by,

kT

Np

kT

Nm

B

M oBoo

33

222 µµµµχ === (14)

where k is Boltzmann's constant and N is the number of Fe3+ ions per unit volume, p is the

magneton number defined in Appendix A, μB is the Bohr magneton and m = pμB. The 1/T

dependence of χMn is known as Curie's Law.

The above theory assumes that the magnetic field acting on each ion is just the applied field B;

field and contributions due to neighboring magnetic ions are neglected. For dilute paramagnetic

materials these other contributions are very small and the approximation is valid. This is not so

for concentrated magnetic materials and ferromagnets.

Apparatus:

1. Adjustable electromagnet with pole pieces

2. Constant power supply (0-16 V, 5A DC)

3. Digital Gauss meter

4. Hall probe for magnetic strength measurement

5. Traveling Microscope

6. Quincke’s tube (an U tube)

7. Measuring cylinder (100ml), Pipette (5ml)/dropper, Wash bottle

8. Specific gravity bottle (25cc)

9. FeCl3for making solutions

10. Electronic balance (Least count = 0.01gm)

11. Connecting cords

Experimental set-up

A schematic diagram of Quinck’s method is

shown in Fig 1. Quinck’s tube is U shaped

glass tube. One arm of the tube is placed

between the pole-pieces of an electromagnet

shown as N-S such that the meniscus of the

liquid lies symmetrically between N-S. The

Fig. 1: Schematics of the set up

length of the limb is sufficient as to keep the other lower extreme end of this limb well

outside the field H of the magnet. The rise or fall h is measured by means of a traveling

microscope of least count of the order of 10-3

cm. The picture of the actual set up is given in

Fig. 2.

Procedure

1. Prepare the FeCl3 solution of known mass (10 or 20gm) in 100 ml water.

2. Calculate the number of moles of Fe3+ ions per unit volume of the solution. 1 mole of a

substance has a weight in grams equal to its molar weight, Wm. The molecular weight is

found by adding up the atomic weights of the constituent atoms of the molecule. If X grams

of FeCl3were dissolved in V m3 of the solution, the number of moles is X/Wm. Each mole

contains NA (Avagadro’s number) of molecules. Thus the number of molecules in V m3 is

N = NAX/Wm

3. Measure the density ρ of your solution using a specific gravity bottle. The method here is to

(a) weigh the bottle + stopper when it is dry and empty

(b) fill it with distilled water and weigh it again

(c) dry it with a dryer and fill it with your solution and weigh it again. The density ρ may be

found, knowing the density of water ρwater, from the following equation

ab

acwater −

−= ρρ (16)

4. Adjust the pole pieces so that the gap between them is about 10 mm (Diameter of the U-

tube is about 8mm).

Fig. 2 Experimental set up for Quicke’s method

5. Connect the electromagnet coils in series to the power supply and ammeter. The field

between the pole pieces must be calibrated as a function of current over an appropriate

range (1- 4A). The magnet may run continuously with a current of 5A (for precautions,

we would avoid prolonged use at 5A, hence the range up to 4A) and for short periods

with 10A. The Hall probe will be used to measure the magnetic field B (how does this

work?). Position the Hall probe using the stand provided so that the same position is

maintained throughout the calibration. With the U-tube removed, insert the Hall probe

into the field region between the flats of the pole pieces.

6. Switch on the Gauss meter and rotate the zero adjustment knob till you get zero reading

on it. Now, switch on the power supply and adjust the current at 1A. Adjust the probe’s

position and orientation until it registers a maximum positive field. Clamp the probe

handle firmly in place so that it cannot move. Measure the flux density B. Slowly

increase the current (I) in small steps and record corresponding values of B.

7. If you record your calibration data with sufficiently small increments of current this will

provide the best definition of the entire curve, which will be linear in a certain range for

smaller values of current and then the slope will decrease as magnetic saturation occurs in

the material of the pole pieces. Note there may also be some magnetic hysteresis present and

for a given current, the field may be slightly different, depending on whether the current is

increasing or decreasing. The magnetic saturation means that the highest values of current do

not produce an equivalent increase in the values of the magnetic field.

8. Bring the current in the power supply back to zero and switch off supply after you finish

calibration.

9. Transfer some of the prepared solution to the U- tube so that the meniscus is at the centre

of the pole-pieces. Focus the travelling microscope on the meniscus and note down the

initial reading for B = 0.

10. Switch on the supply and slowly vary the current up to 4A in steps of 0.5A. The solution

in the tube rises up. Note down the corresponding height of the liquid column for each

value of current.

Observations

Table 1: Data for calibration

Sl# I (A) B (Gauss)

Table 2: Measurement of ρ

Wt. of empty specific gravity bottle (a) = ..

Wt. of specific gravity bottle filled with test liquid (b) = ..

Wt. of specific gravity bottle filled with distilled water (c) = ..

ab

acwater −

−= ρρ

Use ρwater = 1000 kg/m3

Table2. Measurement of h ~ B

Sl# I B B2

Meniscus Reading Difference

h= (b-a)

B=0

(a)

B≠0

(b)

M.S V.S T.R M.S V.S T.R.

Graph:

Plot a graph h~B2 and do straight line fitting to determine slope.

Calculations: Use aχ ~0 and ρa = 1.29 kg/m3

Volume susceptibility, 61004.9 −×−=waterχ

Feχ = …. Feχ ′ = …. Feχ ′′ = ….. C = …… µ = ….

Conclusion and discussion:

Precautions:

1. Scrupulous cleanliness of the U-tube is essential. Thoroughly clean the tube and rinse it

well with distilled water before starting and dry it.

2. Make several sets of measurements to ensure consistency; false readings can arise from

liquid running down the tube or sticking to the sides.

3. Carefully swab down the inside of the U- tube with a cotton bud, to ensure that there are no

droplets of liquid which might interfere with the plastic spacers on the rod which measures

the height of the meniscus.

4. Do not use the U-tube for longer than one laboratory period without recleaning. After

cleaning ask the laboratory technician to dry the tube for you with compressed air.

5. Try to avoid the backlash error of the travelling microscope. The small change of height

may cause you more error in the calculation.

References

[1] I. S. Grant and W.R. Phillips, “Electromagnetism”, (Wiley)

[2]http://www.agoenvironmental.com/sites/default/files/pdf/diamagnetic%20element%

20list.pdf

Appendix A: Magnetic moment values

The magnetic susceptibility of a substance is related to the magnetic dipole moments of its

individual atoms or ions. The total angular momentum of an atom or ion arises from both the

orbital motion and the spin of the electrons. The magnetic dipole moment can be expressed in

the form

m = pμB,

where p, the magneton number, is the dipole moment in units of the quantity μB, which is

known as the Bohr magneton. The Bohr magneton is the atomic unit of magnetic moment

defined by,

μB = eh / 4πme

where, in this equation, e and me are the electronic charge and mass and h is Planck's

constant. The dimensionless magneton number p is usually between 1 and 10 for atomic

systems.

The rules for calculating p can be summarised as follows,

(i) the unfilled electron shells for any atom or ion can be found in standard tables.

(ii) the quantum numbers of the individual electrons can be added

∑=i

ilL and ∑=i

isS

to give the largest values of L and S consistent with the Pauli Exclusion Principle

(iii) the total quantum number J can be found from

J = L - S first half of the electron shell

J = L + S second half of the electron shell

(iv) the magneton number p is given by,

)1( += JJgp where g the so called Landé splitting factor

++−+

+=)1(2

)1()1(

2

3

JJ

LLSSg

takes into account that the spin effectively creates twice as much magnetic moment as the

orbital motion.

(v) the result of these calculations are tabulated in nost textbooks on condensed matter

physics, See the Table 1.

Table 1 Magneton numbers p for some transition metals (TM2+

free ions)

No of electrons

in 3d shell

Ion S L J P

0 Ca2+

0 0 0 0

1 Sc2+

½ 2 3/2 1.55

2 Ti2+

1 3 2 1.63

3 V2+

3/2 3 3/2 0.77

4 Cr2+

2 2 0 0

5 Mn2+

5/2 0 5/2 5.92

6 Fe2+

2 2 4 6.71

7 Fe3+

5/2 0 5/2 5.92

8 Co2+

3/2 3 9/2 6.63

9 Ni2+

1 3 4 5.59

10 Cu2+

½ 2 5/2 3.55

11 Zn2+

0 0 0 0